We present recent results on the critical and pseudo-critical temperatures in
$(2+1)$-flavor QCD with a physical strange quark mass and two degenerate
light quark masses extrapolated to the chiral limit and tuned to the
physical value, respectively. We furthermore discuss implication of the
observed low chiral phase transition temperature, $T_c^0=132_{-6}^{+3}$~MeV,
for the structure of cumulants of conserved charge fluctuations at
vanishing baryon chemical potential and consequences for the possible location
of the QCD critical endpoint in the QCD phase diagram at non-zero baryon
chemical potential.